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Milliken–Taylor Theorem
In mathematics, the Milliken–Taylor theorem in combinatorics is a generalization of both Ramsey's theorem and Hindman's theorem. It is named after Keith Milliken and Alan D. Taylor. Let \mathcal_f(\mathbb) denote the set of finite subsets of \mathbb, and define a partial order on \mathcal_f(\mathbb) by α<β max α |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ''ad hoc'' solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics is gra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ramsey's Theorem
In combinatorics, Ramsey's theorem, in one of its graph-theoretic forms, states that one will find monochromatic cliques in any edge labelling (with colours) of a sufficiently large complete graph. To demonstrate the theorem for two colours (say, blue and red), let and be any two positive integers. Ramsey's theorem states that there exists a least positive integer for which every blue-red edge colouring of the complete graph on vertices contains a blue clique on vertices or a red clique on vertices. (Here signifies an integer that depends on both and .) Ramsey's theorem is a foundational result in combinatorics. The first version of this result was proved by F. P. Ramsey. This initiated the combinatorial theory now called Ramsey theory, that seeks regularity amid disorder: general conditions for the existence of substructures with regular properties. In this application it is a question of the existence of ''monochromatic subsets'', that is, subsets of connected edges of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
IP Set
In mathematics, an IP set is a set of natural numbers which contains all finite sums of some infinite set. The finite sums of a set ''D'' of natural numbers are all those numbers that can be obtained by adding up the elements of some finite nonempty subset of ''D''. The set of all finite sums over ''D'' is often denoted as FS(''D''). Slightly more generally, for a sequence of natural numbers (''n''i), one can consider the set of finite sums FS((''n''i)), consisting of the sums of all finite length subsequences of (''n''i). A set ''A'' of natural numbers is an IP set if there exists an infinite set ''D'' such that FS(''D'') is a subset of ''A''. Equivalently, one may require that ''A'' contains all finite sums FS((''n''i)) of a sequence (''n''i). Some authors give a slightly different definition of IP sets: They require that FS(''D'') equal ''A'' instead of just being a subset. The term IP set was coined by Hillel Furstenberg and Benjamin Weiss to abbreviate "infinite-dimensional ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alan D
Alan may refer to: People *Alan (surname), an English and Turkish surname *Alan (given name), an English given name **List of people with given name Alan ''Following are people commonly referred to solely by "Alan" or by a homonymous name.'' *Alan (Chinese singer) (born 1987), female Chinese singer of Tibetan ethnicity, active in both China and Japan *Alan (Mexican singer) (born 1973), Mexican singer and actor * Alan (wrestler) (born 1975), a.k.a. Gato Eveready, who wrestles in Asistencia Asesoría y Administración *Alan (footballer, born 1979) (Alan Osório da Costa Silva), Brazilian footballer *Alan (footballer, born 1998) (Alan Cardoso de Andrade), Brazilian footballer *Alan I, King of Brittany (died 907), "the Great" *Alan II, Duke of Brittany (c. 900–952) *Alan III, Duke of Brittany(997–1040) *Alan IV, Duke of Brittany (c. 1063–1119), a.k.a. Alan Fergant ("the Younger" in Breton language) *Alan of Tewkesbury, 12th century abbott *Alan of Lynn (c. 1348–1423), 15th cen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
If And Only If
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, ''P if and only if Q'' means that ''P'' is true whenever ''Q'' is true, and the only case in which ''P'' is true is if ''Q'' is also true, whereas in the case of ''P if Q'', there could be other scenarios where ''P'' is true and ''Q'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partition Regular
In combinatorics, a branch of mathematics, partition regularity is one notion of largeness for a collection of sets. Given a set X, a collection of subsets \mathbb \subset \mathcal(X) is called ''partition regular'' if every set ''A'' in the collection has the property that, no matter how ''A'' is partitioned into finitely many subsets, at least one of the subsets will also belong to the collection. That is, for any A \in \mathbb, and any finite partition A = C_1 \cup C_2 \cup \cdots \cup C_n, there exists an ''i'' ≤ ''n'', such that C_i belongs to \mathbb. Ramsey theory is sometimes characterized as the study of which collections \mathbb are partition regular. Examples * the collection of all infinite subsets of an infinite set ''X'' is a prototypical example. In this case partition regularity asserts that every finite partition of an infinite set has an infinite cell (i.e. the infinite pigeonhole principle.) * sets with positive upper density in \mathbb: the ''u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Combinatorial Theory
The ''Journal of Combinatorial Theory'', Series A and Series B, are mathematical journals specializing in combinatorics and related areas. They are published by Elsevier. ''Series A'' is concerned primarily with structures, designs, and applications of combinatorics. ''Series B'' is concerned primarily with graph and matroid theory. The two series are two of the leading journals in the field and are widely known as ''JCTA'' and ''JCTB''. The journal was founded in 1966 by Frank Harary and Gian-Carlo Rota.They are acknowledged on the journals' title pages and Web sites. SeEditorial board of JCTA Originally there was only one journal, which was split into two parts in 1971 as the field grew rapidly. An electronic, [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ramsey Theory
Ramsey theory, named after the British mathematician and philosopher Frank P. Ramsey, is a branch of mathematics that focuses on the appearance of order in a substructure given a structure of a known size. Problems in Ramsey theory typically ask a question of the form: "how big must some structure be to guarantee that a particular property holds?" More specifically, Ron Graham described Ramsey theory as a "branch of combinatorics". Examples A typical result in Ramsey theory starts with some mathematical structure that is then cut into pieces. How big must the original structure be in order to ensure that at least one of the pieces has a given interesting property? This idea can be defined as partition regularity. For example, consider a complete graph of order ''n''; that is, there are ''n'' vertices and each vertex is connected to every other vertex by an edge. A complete graph of order 3 is called a triangle. Now colour each edge either red or blue. How large must ''n'' be in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
